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Weierstrass points on curves have been extensively studied, in connection with many problems. For example, the moduli space Mg has been stratified with subvarieties whose points are isomorphism classes of curves with particular Weierstrass points. For more deatails, we refer for example [3], [6]. At first, the theory of the Weierstrass points was developed only for smooth curves, and for their canonical divisors. In the last years, starting from some papers by R. Lax and C. Widland (see [10], [11], [12], [13], [14], [18]), the theory has been reformulated for Gorenstein curves, where the invertible dualizing sheaf substitutes the canonical sheaf. In this contest, the singular points of a Gorenstein curve are always Weierstrass points. In [16], R. Notari developed a technique to compute the Weierstrass gap sequence at a given point, no matter if it is simple or singular, on a plane curve, with respect 0 to any linear system V ⊆ H C,OC(n) . This technique can be useful to construct
examples of curves with Weierstrass points of given weight, or to look for conditions for a sequence to be a Weierstrass gap Sequence. He used this technique to compute the Weierstrass gap sequence at a point of particular curves and of families of quintic curves. In this paper, we compute the 1-gap sequence of the 1-Weierstrass points on smooth non-hyperelliptic algebraic curves of genus 10 which can be embedded holo- morphically and effectively in algebraic curves of degree 6. Furthermore, the geome- try of Weierstrass points is classified as flexes, sextactic and tentactic points. On the other hand, we show that a smooth non-hyperelliptic curve of genus 10 has no 4-flex, 1, 2-sextactic or 9-tentactic points. Also, an upper bound of the numbers of flexes, sextactic and tentactic points on such curves is estimated.

1 Preliminaries

Notations. We assume the following notations throughout the present paper.

I(C1,C2; p); the intersection number of the curves C1 and C2 at the point p [4].

(q) Gp (Q); the q-gap sequence of the point p with respect to the linear system Q [1, 2].

ω(q)(p); the q-weight of the point p [15].

N (q)(C); the number of q-Weierstrass points on C [8].

2 Q − (ℓ.p) ; the set of divisors in the linear system Q with multiplicity at least ℓ
at the point p [15].

Lemma 1.1. [9, 15] Let X be a smooth projective plane curve of genus g. The number of q-Weierstrass points N (q)(C), counted with their q-weights, is given by

g(g2 − 1), if q =1 (q)  N (C)=  (2q − 1)2(g − 1)2g, if q ≥ 2.  In particular, for smooth projective plane sextic (i.e. g = 10), the number of 1- Weierstrass points is 990 counted with their weights.

(1) Theorem 1.2. [15] Let X be a non-hyperelliptic curve of genus ≥ 3. Write Gp (Q)=

{n1 < n2 < ... < ng}, then

(1) n1 =1,

(2) nr ≤ 2r − 2 for every r ≥ 2, (g − 1)(g − 2) (3) ω(1)(p) ≤ . 2 (4) There are at least 2g + 6 1-Weierstrass points on X.

Remark 1.3. For more details on q-Weierstrass points on Riemann surfaces, we refer for example to [15, 5].

2 Main results

Let X be a smooth projective plane curve of genus 10, and let Q := |K| be its canonical linear system.

9 Proposition 2.1. The linear system Q is g18.

Proof. The result is an immediate consequence, since

dim Q := dim |K| = g − 1=9, and deg (Q) := deg (K)=2(g − 1) = 18.

(1) (1) Corollary 2.2. Let p ∈ X, then ♯Gp (Q)=10 and Gp (Q) ⊂{1, 2, 3, ..., 19}.

9 Lemma 2.3. The set of cubic divisors on X form a linear system which is g18.

3 2.1 Flexes

Definition 2.4. [1, 17] A point p on a smooth plane curve C is said to be a flex point if the tangent line Lp meets C at p with contact order Ip(C, Lp; p) at least three. We say that p is i-flex, if Ip(C, Lp) − 2= i. The positive integer i is called the flex order of p.

Our main results for this part are the following.

Theorem 2.5. Let p be a flex point on a smooth projective non-hyperelliptic plane curve C of g = 10. Let Lp be the tangent lint to C at p such that I(C, Ep)= µf . Then (1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}. Moreover, the geometry of such points is given by the following table:

(1) ωp (Q) Gp(Q) Geometry 2 {1,2,3,...,8,10,11} 1-flex 15 {1,2,3,5,6,7,9,10,13,14} 2-flex 28 {1,2,3,6,7,8,11,12,16,17} 3-flex

Proof. The dimensions of Q(−1.p) and Q(−2.p) do not depend on whether p is 1- Weierstrass point or not, i.e., (1) 1, 2 ∈ Gp (Q).

The spaces Q(−3.p)= ... = Q(−µf .p), consists of divisor of cubic curves of the form

LpR, where R is an arbitrary conic. Hence, dim Q(−ℓ.p) =6 for ℓ = 3, ..., µf . That is, (1) 3 ∈ Gp (Q).

The space Q − (1+ µf ).p consists of divisor of cubic curves of the form LpR, where
R is a conic passing through p. Hence, dim Q − (1 + µf ).p =5. That is,
(1) 1+ µf ∈ Gp (Q).

The space Q −(2+µf ).p consists of divisor of cubic curves of the form LpR, where R
is a conic passing through p with contact order at least 2. Hence, dim Q −(2+µf ).p =
4. That is, (1) 2+ µf ∈ Gp (Q).

The spaces Q − (3 + µf ).p = ... = Q − 2µf .p consists of divisor of cubic curves

of the form L2H, where H is an arbitrary hyperplane. Hence, dim Q − ℓ.p = 3 for p
ℓ =3+ µf , ..., 2µf . That is, (1) 3+ µf ∈ Gp (Q).

4 2 The space Q −(2µf +1).p consists of divisor of cubic curves of the form L H, where
p H is a hyperplane through p. Hence, dim Q − (2µf + 1).p = 2. That is,
(1) 2µf +1 ∈ Gp (Q).

The spaces Q − (2µf + 2).p = ... = Q − 3µf .p consists of divisor of cubic curves 2

of the form LpH, where H is a hyperplane through p with contact order at least 2.

Hence, dim Q − (2µf + 2).p =1. That is,
(1) 2µf +2 ∈ Gp (Q).

3 The space Q − (3µf + 1).p consists of divisor of the cubed tangent line L . Hence,
p dim Q − (3µf + 1).p =0. That is,
(1) 3µf +1 ∈ Gp (Q).

The spaces Q − ℓ.p = φ for ℓ ≥ 3µf +2. Consequently,
(1) 3µf +2 ∈ Gp (Q).

Consequently,

(1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}.

Finally, it is well known that curves of genus 10 can be embedded into algebraic curves of degree 6. Hence, by the famous Bezout’s theorem, the tangent line meets C at the

flex point p with 3 ≤ µf ≤ 6. On the other hand, by Theorem 1.2, it follows that,

µf =66 . Which completes the proof.

Corollary 2.6. If a smooth projective curve X of g = 10 has 4-flex points, then X is hyperelliptic.

Corollary 2.7. On a smooth non-hyperelliptics projective plane curve C of g = 10, the 1-weight of a flex point is given by

(1) ωp (Q)=13µf − 37, where µf is the multiplicity of the tangent line Lp to C at p.

Proof. Let p be a flex point on C, then by Theorem 2.5

(1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}.

5 Consequently,

(1) g ωp (Q) := Σr=0(nr − r)

= (1+ µf − 4)+(2+ µf − 5)+(3+ µf − 6)+(2µf +1 − 7)+(2µf +2 − 8)

+ (3µf +1 − 9)+(3µf +2 − 10)

= 13µf − 37.

(q) Notation. Let Fi (C) be the set of i-flex points which are q-Weierstrass points on (q) (q) C. Also, let NFi (C) denotes the cardinality of Fi (C).

Corollary 2.8. For a smooth non-hyperelliptic projective plane curve C of g = 10, (1) the maximal cardinality of Fi is given by the following table:

(1) i Maximum NFi (C) 1 495 2 66 3 35 4 0

Proof. The number of the 1-Weierstrass points on C is 990 counted with their 1- weight. Hence, 990 NF (1) ≤ [ ], i 13(i + 2) − 37 where i =1, 2, 3.

2.2 Sextactic Points

In analogy with tangent lines and flexes of projective plane curves, one can consider osculating conics and sextactic points in the following way:

Lemma 2.9. [2] Let p be a non-flex point on a smooth projective plane curve X of degree d ≥ 3. Then there is an unique irreducible conic Dp with Ip(X,Dp; p) ≥ 5.

This unique irreducible conic Dp is called the osculating conic of X at p.

Definition 2.10. [1] A non-flex point p on a smooth projective plane curve X is said to be a sextactic point if the osculating conic Dp meets X at p with contact order at least six. A sextactic point p is said to be i-sextactic, if Ip(X,Dp; p) − 5= i. The positive integer i is called the sextactic order.

Now, the main results for this part are the following.

6 Theorem 2.11. Let p be a sextactic point on a smooth projective non-hyperelliptic curve C of g = 10. Let Dp be the osculating conic to C at p such that I(C,Dp; p)= µs. (1) Then Gp (Q)= {1, 2, 3, ..., 7, 1+ µs, 2+ µs, 3+ µs}.

Proof. Again, the dimensions of Q(−1.p) and Q(−2.p) do not depend on whether p is 1-Weierstrass point or not, i.e.,

(1) 1, 2 ∈ Gp (Q).

Moreover, let Lp be the tangent line to C at p, then

div LpR ∈ Q(−2.p) − Q(−3.p),
where, R0 is a conic not through p. That is,

(1) 3 ∈ Gp (Q).

Furthermore,

div LpR1 ∈ Q(−3.p) − Q(−4.p),
where, R1 is a conic passing through p with multiplicity 1. That is,

(1) 4 ∈ Gp (Q).

Also, 2 div L H0 ∈ Q(−4.p) − Q(−5.p), p
where, H0 is a hyperplane not through p. That is,

(1) 5 ∈ Gp (Q).

Similarly, 2 div L H1 ∈ Q(−5.p) − Q(−6.p), p
where, H1 is a hyperplane passing through p with multiplicity 1. So that,

(1) 6 ∈ Gp (Q).

Now, the spaces Q(−7.p)= ... = Q(−µs.p) consists of divisors of cubic curves of the form DP H, where H is an arbitrary line. Hence,

(1) 7 ∈ Gp (Q).

On the other hand, the space Q − (1 + µs).p consists of divisors of cubic curves of
the form DP H, where H is a hyperplane through p. Consequently,

(1) 1+ µs ∈ Gp (Q).

7 Also, the space Q − (2 + µs).p contains only the cubic divisor DP Lp. Then,
(1) 2+ µs ∈ Gp (Q).

Finally, Q − ℓ.p = φ, for ℓ ≥ 3+ µs. That is,
(1) 3+ µs ∈ Gp (Q).

Which completes the proof.

Corollary 2.12. Let p be a sextactic point on a smooth projective non-hyperelliptic curve C of g = 10. Then, the geometry of such points is given by the following table:

(1) ωp (Q) Gp(Q) Geometry 3 {1,2,3,...,,7,9,10,11} 3-sextactic 6 {1,2,3,...,,7,10,11,12} 4-sextactic 9 {1,2,3,...,,7,11,12,13} 5-sextactic 12 {1,2,3,...,,7,12,13,14} 6-sextactic 15 {1,2,3,...,,7,13,14,15} 7-sextactic

Proof. It follows by Theorem 2.11 and Bezout’s theorem, that Dp meets C at p with

8 ≤ µs ≤ 12. Hence, varying µs produces the last table.

Corollary 2.13. If a smooth projective curve X of g = 10 has 1-sextactic or 2- sextactic points, then X is hyperelliptic.

(q) Notation. Let Si (C) be the set of i-sextactic points which are q-Weierstrass points (q) (q) on C. Also, let NSi (C) denotes the cardinality of the set Si (C).

Corollary 2.14. For a smooth non-hyperelliptic projective curve C of g = 10, the (1) maximal cardinality of Si is given by the following table:

(1) i Maximum NSi (C) 1 0 2 0 3 330 4 165 5 110 6 82 7 66

8 Proof. The number of the 1-Weierstrass points of such curves is 990. Hence, 990 NS(1) ≤ [ ], i 3(i + 5) − 21 where i =3, 4, 5, 6, 7. Corollary 2.15. On a smooth non-hyperelliptic projective plane curve C of g = 10, the 1-weight of a sextactic point is given by

(1) ωp (Q)=3µs − 21, where µs is the multiplicity of the osculating conic Dp at p. Proof. If p is a sextactic point on C, then

(1) Gp (Q)= {1, 2, 3, ..., 7, 1+ µs, 2+ µs, 3+ µs}. Consequently,

(1) g ωp (Q) := Σr=0(nr − r)

= (1+ µs − 8)+(2+ µs − 9)+(3+ µs − 10)

= 3µs − 21.

2.3 Tentactic points

Definition 2.16. [1, 7] A point p on a smooth plane curve C of genus g ≥ 2, which is neither flex nor sextactic point, is said to be a tentactic point, if there a cubic Ep which meets C at p with contact order at least 10. The positive integer t such that i := I(C, Ep; p) − 9 is called the tentactic order of p. Moreover, the point p is said to be i-tentactic. Theorem 2.17. Let p be a tentactic point on a smooth projective non-hyperelliptic curve C of g = 10 and let Ep be its osculating cubic curve such that I(C, Ep; p)= µt. (1) Then Gp (Q)= {1, 2, 3, ..., 9, 1+ µt}. Moreover, the geometry of such points is given by the following table: (1) ωp (Q) Gp(Q) Geometry 1 {1,2,3,...,9,11} 1-tentactic 2 {1,2,3,...,9,12} 2-tentactic 3 {1,2,3,...,9,13} 3-tentactic 4 {1,2,3,...,9,14} 4-tentactic 5 {1,2,3,...,9,15} 5-tentactic 6 {1,2,3,...,9,16} 6-tentactic 7 {1,2,3,...,9,17} 7-tentactic 8 {1,2,3,...,9,18} 8-tentactic

9 Proof. Since, the point p is neither flex nor sextactic, then

dim Q(−ℓ.p)=9 − ℓ forℓ =1, 2, 3, ..., 9.

Hence, (1) 1, 2, 3, ..., 9 ∈ Gp (Q).

Moreover, assuming that I(C, Ep)= µt, then

div (Ep) ∈ Q(−µt.p) − Q(−(1 + µt).p).

(1) Therefore, 1 + µt ∈ Gp (Q). Consequently,

Gp(Q)= {1, 2, 3, ..., 9, 1+ µt}.

(1) Finally, 19 ∈/ Gp (Q) as n10 ≤ 18.

Corollary 2.18. If a smooth projective curve X of g = 10 has 9-tentactic points, then X is hyperelliptic.

(q) Notation. Let Ti (C) be the set of i-tentactic points which are q-Weierstrass points (q) (q) on C. Also, let NTi (C) denotes the cardinality of the set Ti (C).

Corollary 2.19. For a smooth projective non-hyperelliptic curve C of g = 10, the (1) maximal cardinality of Ti is given by the following table:

(1) i Maximum NTi (C) 1 990 2 495 3 330 4 247 5 198 6 165 7 141 8 123 9 0

Proof. The number of the 1-Weierstrass points of a smooth projective non-hyperelliptic curve of genus g is (g − 1)g(g + 1). Hence, in particular, 990 NT (1) ≤ [ ], i i where i =1, 2, 3, ..., 8.

10 Concluding remarks. We conclude the paper by the following remarks and comments.

• The 1-Weierstrass points of an algebraic curve of g = 10 are classified as flexes, sextactic and tentactic points. In particular, the present authors computed the 1-gap sequences of such points on non-hyperelliptic curves of genus 10. Furthermore the geometry of these points is investigated and an upper bound for their number is estimated.

• The main theorems constitute a motivation to solve other problems. One of these problems is the investigation of the geometry of the 1-Weierstrass points of Kuribayashi sextic curve with three parameters a, b and c defined by the equation:

6 6 6 3 3 3 3 3 3 KIa,b,c : X + Y + Z + aX Y + bX Z + cY Z .

However, this problem will be the object of a forthcoming work.

References

[1] Alwaleed K., Geometry of 2-Weierstrass points on certain plane curves, Ph. D. Thesis, Saitama Univ., 2010.

[2] Alwaleed K. and Kawasaki M., 2-Weierstrass points of certain plane curves of genus three, Saitama Math. J., 26 (2009), 49-65.

[3] Arbarello E., Weierstrass Points and Moduli of Curves, Compositio Mathematica 29 3 (1974), 325342.

′ [4] Cox, D., Little, J. and O Shea, D., Ideals, Varities and Algorithms, Springer- Verlag, New York, 1992.

[5] Del Centina, A., Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Luckensatz” to the 1970s, Ann Univ. Fer- rara, 54 (2008), 37-59.

[6] Diaz S., Exceptional Weierstrass Points and the Divisor on Moduli Space that they define, Memoirs of the A.M.S., 56, (1985) n. 327.

[7] Farahat M., Geometry of 3-Weierstrass points on genus two curves with extra involutions, Ph. D. Thesis, Saitama Univ., 2012.

[8] Farahat M. and Sakai F., The 3-Weierstrass points on genus two curves with extra involutions, Saitama Math. J., 28 (2011), 1-12.

11 [9] Farkas, H. M. and Kra, I., Riemann Surfaces, GTM71, Springer Verlag, New York, 1980.

[10] Lax R.F., Weierstrass Points of the Universal Curve, Math. Ann. 42 (1975), 3542.

[11] Lax R.F., On the distribution of Weierstrass Points on Singular Curves, Israel J. Math. 57 (1987), 107115.

[12] Lax R.F., Gap Sequences at Singularity, Pacific Journal of Mathematics 150 1 (1991), 111122.

[13] Lax R.F., Widland C., Weierstrass Points on Rational Nodal Curves of Genus 3, Canad. Math. Bull. 30 (1987), 286294.

[14] Lax R.F., Widland C., Weierstrass Points on Gorenstein Curves, Pacific Journal of Mathematics 142 1 (1990), 197208.

[15] Miranda, R., Algebraic Curves and Riemann Surfaces, Amer. Math. Soc., 1995.

[16] Notari R., On the computation of Weierstrass gap sequences, Rend. Sem. Mat. Univ. Pol. Torino 57, 1 (1999).

[17] Vermeulen, A., Weierstrass points of weight two on curves of genus three, Ph. D. Thesis, Amsterdam Univ., 1983.

[18] Widland C., Weiestrass Points on Gorenstein Curves, Ph. D. Thesis, Louisiana State University 1984.

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